wildhaber



Dec. 4, 1928.

E. WILIDHABER GEAR Filed June 10, 1926 4 Sheets-Sheet E. WILDHABER GEAR Filed June 10, 1926 4 Sheets-Sheet 2 E. WILIDHABER GEAR Filed June 10, 192.6 4. Sheets-Sheet 3 INVENTOR Br I iz'ldhaber Dec. 4, 1928. 1,694,028

E. WHLDHABER GEAR Filed June 10, 1926 4 Sheets-Sheet 4 INVENTOR E'rzzeslT iilqmaber the tooth surfaces of the other member of the pair are parts of such surfaces of concave circular cross-section.

Referring now to the drawings, wherein the same reference numerals indicate the same parts in all figures, 10 and 11 are the axes of a pair of gears constructed accord ing to this invention. These axes are dis posed at an angle A to each other. shown, the angle A is different from a right angle. The angle has been so assumed in order to permit of abroad treatment of the invention, but it is obvious that the angle maybe also and, in fact, in the preferred embodiment of the invention is, a right angle. 12 and 13 are respectively surfaces of revolution coaxial with the axes 10 and 11 which may be considered the pitch surfaces of a pair of gears constructed according to this invention.

15 and 16 are lines traced on the pitch surfaces 12 and 13 by the center 17 of a sphere 18 during the motion of the center of the sphere along a line 20. As will presently be explained, the path 20 of the sphere center is so selected that it may be considered the line of contact between the two surfaces 12 and 13. In other words, the line 20 should be a line which, when rotated about the axes 10 and 11 respectively will form two surfaces of revolution contacting along the same line.

The velocity of the generating sphere 18 along the line 20 is such that the lines 15 and 16 traced on the pitch surfaces 12 and 13 by the sphere center 17 continuously contact with each other in the moving center 17. In other words, the pitch lines 15 and 16 have a common tangent at the point 17. This tangent extends in the direction of relative sliding of the two pitch surfaces at the point 17.

Any two surfaces of revolution which are coaxial with the gears and which contact with each other along a line can be used as pitch surfaces for a pair of gears constructed according to the present invention. In the preferred embodim nt of the invention, hyperboloids of revolution are employed for the pitch surfaces, as shown in Figures 1 and 2. Such pitch surfaces are known to contact along a straight line when properly selected.

The sphere 18 moving along the straight line 20 will sweep out tube-like surfaces on the gears. These tube-like surfaces naturally contact with the sphere 18 itself in a section which passes through the sphere center 17 and which is perpendicular to the pitch line 15 or 16. Inasmuch as the lines 15 and 16 have a common tangent at the point 17, that is, have the same direction at this point, the tube-like surfaces produced on the mating gears will contact with the sphere 18 in the same section, that is, in the major circle 19 whose plane is perpendicular to the common tangent. If, therefore, part of the surface swept out by the sphere 18 is embodied as a surface of convex profile on one gear and a corresponding part of ti o surface swept out by the sphere is embodied as a surface of concave profile on the other ear, these surfaces will contact with the imagn sphere 18 and with themselves along the sa line and in mesh they will re roduce t e relative motion which was employed in pr cing them. The gears will, therefore, rotate in the constant ratio of their respective tooth nun! era and hence will transmit uniform motion.

The method by which the path 20 is select al and by which the rate of travel of "he imaginary sphere 18 is ac mined u be explained. Line 20 is assumed in a p .c parallel to the axes 10 and 11. For convenience, 8 use is made of the following symbols s The absolute velocity of the sphere center 1? along line 20.

P The pitch of th g of the line '20, the i the use between two adjacent contact point and 17 or 1'? and 17 ind cated in Figures 1 and G,g The a gles between line 20 and the projected gear and pinion 10 and 11 respectively.

N912 The tooth numbers of gear and pinion respectively.

V,'v The relative velocities of the sphere center 17 with respect to the gear and pinion.

Vs, o, The components of the relative Yelocities V, o the direction the line 20.

Va, o T he components of the relative velocities V, v normal to :20 and in a plane parallel to the axes 10 and 11.

V o The components of the i "ive velocities V, v perpendicular to the We uses 10 and 11 that is perpendicular to the drawing plane of Figure 1.

' IVnu The angular velocities about the axes 10 and 11.

Z,.2 The distances of the line 20 from the axes 10 and 11 respecn. ly.

E 'ihe amount of offset between the axes 10 and 11.

et= he distance of 1 from thepoint 2t l the direction Ina cle of the hyptrboloi the line 20. p

to sin G, M sin g The distances of the spher center 17 free 1 respectively, in pro ure 1.

Assun? 1g the directions of rota cated by the arrows and :27 F1 and 2 that the spl ""c inwardly in the Llcei the components Va, I

velocity V can be determined n'h .me "n methods of kinematics as follows 12:. V8 sZ 1V sin V'n Z 1V cos G Vz a S111 G V] In like manner the con'iponents of the rela- 13o Vn V2 =2;

Vs V2 Substituting for these various components their values as deternnned above We have:

ZW cos G fl r si n GW 2w cos 9 a sin gw or:

p Z and can also be expressed as functions ot the ol'lset E. Y

z tan 9 tan g tan G If the axes 10 and 11 are disposed at right i and the last angles as is usual, tan 9 two equations become:

Ecpiations I or ll determine th 100:. tion of the line 20 so that this line is'a contact line betweentwo hyperboloids having axes 10 and 11.

With the values obtained above equation (2) can be trz-insitorined to:

8+ 2w sin g u sin gw and to:

s s i 2 VJ sin G w sin g s i the movement oi the center 1'? per revolution divided by the corresponding angular motion 271', that is,

velocity 8 of the center 1?.

Substituting these values in the last equation, we have 1' the axes 10 and 11 are disposed at right angles, G SO g and sin-(i= cos y. p

Equation (11) .detern'iincs the traveling H the center 1'? moves by an amount P per tooth as indie ate-d above on a line 20 Whose location is given in equation (I) thenit will trace two sets of curves 15 and 16 which are continuously tangent to each other andwhich, therefore, fulfill the requirements of the pitch lines as previously explained.

It the gear is assumed to turn opp sitely to the arrow 26, the required pinion will be otopposite hand and in formula (11) the negative sign will be replaced by a positive sign. i

The pitch 1 along line is constant and therefore, the center 17 off the sphere 18 moves at a constant rate along line 20.

In the case of hypoid gears tapered por tions of the pitch surfaces are used as indicated in Figures 1 and 2. the diameter of the sphere 1 8 is then taken preferably 1 to 3 tii'nes the normal pitch of the teeth. In the case or WOIIII gee rs, the portions the pitch surfaces used are near the gorge circles. as shown in Figure 3. Either ln'anch 20 or 520" of the contact line of the hyperboloids 80 and 3111121 be used as the path of ter 17. For Worni gears, the Clltilll of the basic sphere 18 may be larger than for hypoids. V faces are ennployed as-tooth surfaces which are free from interference. in many case and especially in Worm gears. the surfaces andSl Whieh have been called the pitcl surfaces Will lie entirely outside oi or inside of the actual teeth. 7

In the embodiment snown in ITEQIUI'C' l. the

pitch surface 10 is outside the actitaal. teeth or threads. The tooth surfaces surfaces of concave profile such as might be (?D.Vtli11 )l by a sphere 11 which moves at a constant rate e sphere cen- Such parts of the conjugate Sim tit) line 42 are indicated at 44, 44, and 4 In these positions contact between the sphere and the enveloped tooth surface takes place along the major circles 45, 45 and 45". Only the halves of these great circles are shown and only a small part of these circles are used on the tooth surfaces, as indicated at 46, 46', and 46.

The preferred method of producing the worm 47 whose pitch surface has been indicated 211 in Figure 4, is illustrated diagrammatically in Figure 5. Cutting tools'48 are en'iployed which have a cutting surface covering or representing part of the spherical surface 41. The radius 49 of the cutting surface of the tool is equal to the radius of the basic sphere 41.. The cutting tool is rotated on its axis while the blank rotates on its axis 43 and simultaneously a relative movement is imparted between the tool and blank along the generatrix 42.. This relative movement-is preferably at a constant rate and is in timed relation to the blank rotation. Three different positions 48, 48, and 48 of the cutting wheel are shown in the figure. The contact between the cutting surface of the wheel and the finished tooth surfacetakes place along the lines 46, 46 and 46 which are parts of the major circles 45, and 45 shown in Figure 4.

Instead of a disc shaped cutting wheel 48, any other suitable type of cutting or grinding wheel may be used, the cutting wheel 50 of the face mill type shown in Figure (3. The tools have spherical cutting surfaces of the same diameter as the sphere 41.v

The preferred method of producing a worm gear to mate with the worm 47 is illustrated diagrammatically in Figure 7. 51 indicates tl ieral contour of the worm wheel and is axis. A cutting or grinding wheel having a concave cutting surface or edge is employed. The cutting surface of this wheel may be part of a concave or inside spherical surface but in order to avoid interference ordinarily such a surface is not used. Instead. preferably, the cutting wheel is provided with a cutting surface of concave circular profile which is part of asurface of revolution whose center is disposed outside of the, axis of the tool and preferably also outwardly of its profile. In order that such a cutting surface may be able to contact with the blank along the arcs 46, 46 and 46, the cutter must be swivelled about an axis 54 which continuously p asses through the sphere center 44 as the sphere center moves along the gen-eratrix 42. Three different positions of the tool 5-3 are shown at 53, 53' and 53" in Figure 7. The positions of the tool axis are indicated at 55, 5 5 and while the positions of the axis 54 about which the tool is swivelled, and which continuously passes through the sphere centers 44, 44 and 44, are indicated at 54, 54' and 54". The plane containing the cutter axis 55 and the sphere center 44 is identical with the plane of the major circle 45 along which the contact between the basic sphere and the tooth surface of the worm wheel is effected. During the relative movement of the tool and blank along the generatrix 42, the blank is rotated on its axis 52 in timed relation. A swivelling move nent of the character described can be readily efiected.

The preferred method of producing a hypoid pinion or gear is illustrated diagrammatically in Figure 8. The axis of the pinion whose pitchsurface is indicated at 61 is supposed to be inclined by an angle 9 to the drawing plane. The generatrix 62 of the pitch surface 61 lies, therefore, in the plane of the drawing. A tooth surface of the pinion is a surface enveloped by a sphere 63 whose center 64 moves preferably at a constant rate along the generatrix 62. Contact between the imaginary sphere 63 and the finished tooth surface takes place along a major circle 66 whose plane is perpendicular to the pitch line 67 of the pinion. This plane is identical with'the .plane containing the sphere center 64 and the axis of the cutting or grinding wheel 68 which is used to sweep out the pinion tooth surfaces. The movements imparted between the tool 68 and the blank are those employed in producing the worm. wheel 51. The normal to the plane of the major circle 66 projects into a line 69. The location of this normal which determines the spiral angle of the pinion teeth re uired to obtain the desired tooth contact can e determined from the angle m which the no an al makes with the generatrix 62 and from the inclination z' of this normal to the plane of Figure 8. The angle on can be computed like the angle of relative velocity o.

7073 equals therefore Pu 2w sin 9 which is a constant. The distance 78- 1 itself equals a. Point 70 can be located, there- 1 p 1 fore, by plotting on hno 72 a distance 27F Sin- 9 from the point 74. Anyline 71, 71 which is perpendicular to the normal (39, 69 can then meanes tan 9 Angle -i can be found graphically asfollows. The constant distance 271'2 cos y is plotted on line 62 from point 6st to 76. The line 77 is drawn through 76 perpendicular to line 62 to intersect line 69 in point 78. A unit, such as one inch, is plotted on a line drawn through 78 perpendicular to 69, that is parallel to 71. The line connecting the resulting point 79 with 64f and shown in dotted lines at 80 includes the angle 2' with the line 69. The tangent of this angle fulfills the above equation.

From the equations last derived in the spiral angle of the pinion can be computed so that the gear and pinion will mesh along the contact line 62. F

A pair of hypoid gears constructed according to the preferred form of this invention are illustrated in. l igure 9. 'fhe pinion axis 8]. is offset from the gear axis .82 by the distance ll and the pinion is provided with teeth 83 whose side surfaces are of convex profile +tan g in planes perpendicular to the pitch lines of the pinion, while the gear is provided with teeth 8% whose side surfaces are of concave profile in such, planes. The spiral angle of the pinion teeth is preferably made larger than the spiral angle of the gear teeth. By such a construction and by determining the genciutrices of the pitch surfaces of gear and pinion so that the anglebetween the generatriz; and a line drawn parallel to the ZLXlS of n the blank is larger than p, Where tan 29 gle of the pinion or smaller member of the pair is larger than the spiral angle of the gear or larger member of the pair the danger of interference is much smaller on the longitudinally concave sides 85 of the pinion teeth, than on the longitudinally convex sides 86 of the teeth. For this reason the pressure angles are preferably made different on the two sides of the teeth. The pressure angle will be smaller on the concave sides 85 of the teeth in the embodiment considered. Frequently the longitudinally concave sides of the pinion teeth maybe made the driving sides and usually the pressure angle onthis side of the teeth maybe made as low as zero degrees without causing, undue interference a feature which could never be achieved ongears with intersecting or parallel axes. The pressure angles of the opposite sides 86 of the teeth may be correspondingly increased over the corresponding sides of the teeth of spiral bevel gears. p Figures 10 and 11 are corresponding sections through the pitch surfaces of a hypoid gearv and pinion respectively taken in planes perpendicular to the pitch lines of the two members. For convenience the spacing of the profiles is shown constant in the drawing;

Both sides of the teeth 84 and 83 of the gear and pinion are surfaces such as might be enveloped by a sphere. The sides 87 and 88 of the gear teeth 84 are of substantially concave circular profile, while the sides 85 and 86 of the pinion teeth 83ers of substantially convex circular profile. To secure the desired different pressure angles on the opposite sides of the teeth, the two sides 85 and 86 of the pinion teeth are constructed to correspond to different pitch surfaces 90 and 91 the tooth profiles 85 and 86 being substantially circular arcs of radii 92 and 93 respectively whose centors are at 94c and 95 respectively on the pitch surfaces 90 and 91 respectively. This struc-' ture is possible because the pitchsurfaces of hypoid gears do not depend solely on the rela tive position of a pair of gears and their ratio as the pitch surfaces of bevel and spur gears. On the contrary an infinite number of pairs of pitch surfaces can be assumed for a given position of a pair of hypoid gears of a given ratio. This is evident also from the formulas already derived. Itshould be noted that the average pressure angle of profiles 85 is below the usual pressure angle of gears and is around zero degrees.

The pressure angles of the two sides of the gear teeth are made differentto conform to those of the mating sides of the pinion teeth. On account of the different pressure angles on the two sides of the teeth the central lines of the teeth are inclined at an angle to the radii of the gea as indicated in Figure 11, where the center line 96 of the pinion teeth is shown inclined at an angle 97 to the. radius 98. This offsets any und rcut and avoids interference in contradistinction of such a structure in other types of gears.

Where the longitudinally concave sides 99 of the pinion teeth. and the mating longitudinally convex sides 100 of the gear teeth are used as the driving sides, the opposite sides of the gear and pinion teeth need not beconstructed according to this invention but may be formed in any suitable Way. A section thron -h a pair of hypoid gears having a modified profile on one side of the teeth is shown in Figure 12. The profiles of the driving sides of the pinion teeth are convex and the profiles of the longitudinally convex sides of the gear teeth are concave, but the 0 )posite sides 101 and 102 of the gear and pinion teeth are both convex. Here again the pressure angles are preferably made different on the two sides of the teeth, the pressure angle of the profiles 99 being considerably below the usual pres sure angle of gears and around zero degrees. The profiles of the tooth sides 99 and 100 are circular arcs of equal radii 103 the center of a pair of such tooth profiles being shown at 104 on the pitch surfaces 105 and 106. The tooth sides 101 and 102 with the larger pressure angles may be formed in any suitable way.

I The tooth shape of gears constructed according to this invention can be applied also to a pair consisting of a crown gear and a worm, where the crowngear has a plane pitch surface and the worm has a cylindrical pitch surface. In this case the sphere center moves along the straight line of contact between said pitch plane and the pitch cylinder and the lines traced by the sphere center on the two pitch surfaces are preferably identical with the pitch lines produced on the hob and gear described in my copendi'ng application, Serial No. 38,724, filed June 22, In development the pitch lines of the worm are then parabolas.

Gears constructed according to this invention can be produced quickly and in a sinr ple operation, the blanks being cut .withou roll and preferably in a continuous indexing process.

While I have described and illustrated cer tain preferred embodiments of my invention, it will be understood that the invention is capable of further modification within its scope and within the limits of the appended claims and that this application is intended to cover any variations, uses, or adaptations, of this invention following in general, the principles of the invention and including such departures from the present disclosure as come within known or customary practice in the gear art and as may be applied to the essential features hereinbefore set forth and as fall within the limits of the appended claims.

Having thus described my invention, what I claim is:

1. A pair of gears adapted to mesh with axes non-intersecting and non-parallel, each of which has teeth extending across its face along lines inclined to the generatrices of its pitch surface, the profiles of'the side tooth surfaces of said in planes perpendicular to their respective pitch lines being sin gle circular arcs.

2. A pair of gears adapted to mesh with axes non-intersecting and non-parallel, each of which has teeth extending across its face pitch surface, said gears having side tooth surfaces whose profiles, in planes perpendicular to their respective pitch lines, are in the form of single circular arcs having centers located on the pitch surfaces of the respective gears.

4:. A pair of gears adapted to mesh witl axes non-intersecting non-parallel, each of which has teeth extending across its face alonglines inclined to the generatrices of its pitch surface, said gears having side tooth surfaces whose working portions in planes perpendicular to their respective pitch lines are in the form of single circular arcs having centers which lie outside the respective teeth and on the pitch surfaces of the respective gears.

5. A pair of gears adapted to mesh with axes non-intersecting and non-parallel, each of which has teeth extending across its face along lines inclined to the generatrices of its pitch surface, one of said gears being provided with active tooth surfaces which in planes perpendicular to its pitch line are exclusively convex circular arcs and the other of said gears having active tooth surfaces which in planes perpendicular to its pitch lines are exclusively concave circular arcs.

6. A pair of gears adapted to mesh with axes non-intersecting and non-parallel, each of which has teeth extending across its face along lines inclined to the generatrices of its pitch surface, said gears having active tooth surfaces, thev profiles of which in planes perpendicular to their respective pitch lines are single circular arcs, mate profiles having suhstantially the same radius.

7. A pair of gears adapted to mesh witl axes non-intersecting and non-parallel, eacl of which has teeth extending across its face along lines inclined to the generatrices of pitch surface, one of said ears having act-ire tooth surfaces whose profiles, in planes perpendicular to its pitch lines, are exclusively convex circular arcs, and the other of said gears having active tooth surfaces whose profiles, in planes perpendicular to its pitch lines are exclusively concave circular arcs, the centers of the profiles of each gear be ing located on the respective pitch surfac 8. A pair of gears adapted to mesh with axes non-intersecting and non-parallel, each of which has teeth extending across its face along lines inclined to the generatrices of its llii tary tooth profiles in planes perpendicular to the pitch lines otthe respective gears, inate profiles being respectively, convex and concave circular arcs oi suhstai'itially equal rad i.

A pair oi gears adapted to niesh with axes non-intersecting and non-parallel, each of which has teeth extending across its face along lines inclined to the generatrices of its pitch surface, said having con'ipleinentary toot-h profiles in planes perpendicu lar to the pitch lines of the respective (rears, mate profiles being respectively convex and concave arcs of circles whose diameters are 1 to 3 times the normal pitch.

ill. A pair of gears adapted to mesh with axes non-intersecting and non-parallel, each of which has teeth attending across its face alon lines inclined to the generatrices of its pitch surface, said. gears having complementary tooth profiles in planes perpendicular to the pitch lines of the respective gears, inate profiles being wholly convex and wholly concave respectively.

11. A pair 01 gears adapted to mesh with axes noi'i-intersecting and non-parallel, each of which has teeth extending acr ss its face along lines inclined to the generatrices off its pitch surface, salt gears having complementai v tooth profiles in planes perpendicular to the pitch lii profiles being respectively convex and concave circiillar arcs, the profiles of opposite sides of the teeth of the respective gears being of unequal radii.

19/. A pair ot gears adapted to mesh with axes non-intersccting and non-parallel, each 0t which has teeth extending across its face along lines inclined. to the generatrices of its pitch surface, said gears having tooth profiles on the driving side of the respective teeth which are complementary in planes perpendicular to the pitch lines of the respective nears, 'inate driving profiles bein respectively convex and concave circular arcs.

13 A pair of gears adapted to mesh with axes non-intersecting and non-parallel each of which has teeth extending across its face along lines inclined to the generatrices of its pitch surface, said gears having tooth profiles on the (lllVll'lg sides of the respective teeth which are complementary in planes perpendicular to the pitch lines of the respective gears, inate driving profiles being wholly convex and wholly concave respectively.

1 1:. A. hypoid gear having teeth extending across i iace along lines inclined to the gene atrices of its pitch surface, said gear hmv teeth whose active tooth surfaces have profiles in the form single circular arcs,

the centers of the profiles ot' a tooth lying.

on. opposite sides oi said. tooth.

generatrices of its pitch surfaces, said gear es of the respective gears, mate having teeth whose active tooth surfaces have profiles in the form of single circular arcs, the centers oi the profiles of a tooth lying on opposite sides of said tooth and being lecatec. substantially on the pitch surface of said generatrices of its pitch surface, said gear" havin teeth which are 01": constant )itch C3 along a straight line offsetitrom its axis, the

profiles of the active surfaces of which are in the form of single circular arcs, the centers 01" the lrofiles of a tooth lying on opposite sides of said tooth.

17. A hypoid gear having teeth extending across its face along lines inclined tothe generatrices of its pitch surface, said gear having teeth whose active tooth surfaces have profiles in the fennel single arcs of circles whose diameters are 1 to 3 times the normal pitch, the centers of the profiles of the tooth lying on opposite sides of said tooth.

18. A pair oft gears adapted to mesh with. axes non-intersecting and non-parallel and each of which has teeth extending across its face along lines inclined to the generatrices of its pitch surface, said gears containing respectively convex and concave mate tooth profiles of substantially the same shape in planes perpendicular to their pitch lines.

19. A pair of adapted to mesh with axesv non-intersecting and non-parallel, each of which has teeth extending across its taco along lines inclined to the generatrices of its pitch surface and each of wiich is of constant pitch along a straight line offset from its axis, said. go rs having coniplenientary tooth profilesin planes perpendicular to their pitch lines, mate profiles being respectively convex and concave circular arcs.

20. A. pair of gears adapted to mesh with axes non-intersecting and non-parallel and each of which has teeth extending across its face along lines inclined. to the genoratrices of its pitch surface, the inclination angle of the teeth of the smaller gear being larger than the inclination angle of the teeth of the larger gear, said gears ha ving complementary tooth profiles in planes perpendicular to their pitch lines, mate profiles being respectively convex and concave circular arcs.

21. A pair of gears adapted to mesh with axes non-intersecting and non-parallel, one of which has tooth surfaces such as might he enveloped by a sphere moving along a straight line otlset from the axes of the blank and inclined to a line parallel to the blank axis by an amount greater than p, where tan n P E and a and N are the tooth numbers of said gear and the mate respectively,

22. A pair. of gears adapted to mesh with axes nonanterscctmg and non-parallel, each of which has teeth extending across the face along lines inclined to the gcneratrices of its pitch surface and each of which is of constant pitch along a straight line oll'sel; from its axis, said gears having, respectively, exclusively convex and concave profiles, inate profiles being complementary.

23. A pair of gears adapted to mesh with axes non-intersecting and non-parallel and each of Which has teeth extending across its face along lines inclined to the gene ratrice's 10 tively, exclusively convex and concave pro- 15 files, mate profiles being complementary.

ERNEST lVILDHABER. 

